Subtraction of Binary Numbers for JSS 1

Subtraction of Numbers in Base 2

Performance Objectives

By the end of the lesson, students should be able to:

1. Subtract two- or three-digit binary numbers.

Meaning of Binary Numbers

Binary numbers, also known as base 2 numbers, are numbers expressed in a system that uses only two digits: 0 and 1.

In the everyday decimal system (base 10), counting is done in groups of tens, using the digits 0 to 9. In the binary system, counting is done in groups of twos. The place values in binary are based on powers of two, starting from the rightmost column: units ($2^0 = 1$), twos ($2^1 = 2$), fours ($2^2 = 4$), and so on. Subtraction in base 2 follows similar place-value rules as base 10, but borrowing is based on groups of two instead of groups of ten.

Rules of Binary Subtraction

Binary subtraction is performed column by column, starting from the right (the units column) and moving to the left. The four basic subtraction facts for binary digits are:

1. $0_2 – 0_2 = 0_2$
2. $1_2 – 0_2 = 1_2$
3. $1_2 – 1_2 = 0_2$
4. $10_2 – 1_2 = 1_2$ (This occurs when $1$ is borrowed from the next column to the left, which represents $2$ in the current column)

When subtracting a larger digit from a smaller digit (such as $0 – 1$), borrowing from the next column to the left is necessary. In base 10, borrowing $1$ from the tens column gives $10$ units. In base 2, borrowing $1$ from the twos column gives $2$ units (written as $10_2$ in binary).

Using Base 2 Blocks and Charts for Subtraction

Base 2 blocks and binary charts help to visualize how subtraction works. In a base 2 block system, different blocks represent different place values:

1. A unit block represents $1$ (the units place, $2^0$).
2. A two-block (two unit blocks joined together) represents $2$ (the twos place, $2^1$).
3. A four-block (four unit blocks joined together) represents $4$ (the fours place, $2^2$).

To subtract using blocks, the initial number is represented with blocks, and the number to be subtracted is removed. If there are not enough unit blocks to remove, a two-block must be broken down into two unit blocks. This represents the process of borrowing.

Observe the diagram below, which illustrates how base 2 blocks are used to subtract $11_2 – 01_2$.

Worked Examples: Two-Digit Binary Subtraction

The following examples demonstrate how to subtract two-digit binary numbers step by step.

Example 1: Subtract $01_2$ from $11_2$.

First, arrange the numbers vertically, aligning the place values:

$$
\begin{array}{rcc}
& \text{Twos} & \text{Units} \\
& 1 & 1_2 \\
– & 0 & 1_2 \\
\hline
& & \\
\hline
\end{array}
$$

The subtraction is performed column by column from right to left:

1. In the units column, subtract the digits: $1 – 1 = 0$. Write $0$ in the units column of the answer.
2. In the twos column, subtract the digits: $1 – 0 = 1$. Write $1$ in the twos column of the answer.

The final vertical arrangement is:

$$
\begin{array}{rcc}
& 1 & 1_2 \\
– & 0 & 1_2 \\
\hline
& 1 & 0_2 \\
\hline
\end{array}
$$

Therefore, $11_2 – 01_2 = 10_2$.

Example 2: Subtract $10_2 – 01_2$.

Arrange the numbers vertically:

$$
\begin{array}{rcc}
& \text{Twos} & \text{Units} \\
& 1 & 0_2 \\
– & 0 & 1_2 \\
\hline
& & \\
\hline
\end{array}
$$

The subtraction is performed as follows:

1. In the units column, the subtraction is $0 – 1$. Since $1$ cannot be subtracted from $0$, borrow $1$ from the twos column.
2. Borrowing $1$ from the twos column leaves $0$ in that column, and transfers $2$ units (written as $10_2$) to the units column.
3. Now, in the units column, perform the subtraction: $10_2 – 1_2 = 1_2$. Write $1$ in the units column of the answer.
4. In the twos column, the remaining digit is $0$. Perform the subtraction: $0 – 0 = 0$. Write $0$ in the twos column of the answer.

The final vertical arrangement is:

$$
\begin{array}{rcc}
& \overset{0}{\cancel{1}} & \overset{10}{0}_2 \\
– & 0 & 1_2 \\
\hline
& 0 & 1_2 \\
\hline
\end{array}
$$

Therefore, $10_2 – 01_2 = 1_2$ (or $01_2$).

Worked Examples: Three-Digit Binary Subtraction

The following examples demonstrate how to subtract three-digit binary numbers, including cases that require multiple borrowing steps.

Example 1: Subtract $101_2$ from $111_2$.

Arrange the numbers vertically, aligning the fours, twos, and units columns:

$$
\begin{array}{rccc}
& \text{Fours} & \text{Twos} & \text{Units} \\
& 1 & 1 & 1_2 \\
– & 1 & 0 & 1_2 \\
\hline
& & & \\
\hline
\end{array}
$$

Perform the subtraction column by column from right to left:

1. In the units column, subtract the digits: $1 – 1 = 0$. Write $0$ in the units column of the answer.
2. In the twos column, subtract the digits: $1 – 0 = 1$. Write $1$ in the twos column of the answer.
3. In the fours column, subtract the digits: $1 – 1 = 0$. Write $0$ in the fours column of the answer.

The final vertical arrangement is:

$$
\begin{array}{rccc}
& 1 & 1 & 1_2 \\
– & 1 & 0 & 1_2 \\
\hline
& 0 & 1 & 0_2 \\
\hline
\end{array}
$$

Therefore, $111_2 – 101_2 = 10_2$ (or $010_2$).

Example 2: Subtract $110_2 – 011_2$.

Arrange the numbers vertically:

$$
\begin{array}{rccc}
& \text{Fours} & \text{Twos} & \text{Units} \\
& 1 & 1 & 0_2 \\
– & 0 & 1 & 1_2 \\
\hline
& & & \\
\hline
\end{array}
$$

Perform the subtraction as follows:

1. In the units column, the subtraction is $0 – 1$. Borrow $1$ from the twos column. This leaves $0$ in the twos column and makes the units column $10_2$.
2. In the units column, subtract: $10_2 – 1_2 = 1_2$. Write $1$ in the units column of the answer.
3. In the twos column, the remaining digit is now $0$. The subtraction is $0 – 1$. Borrow $1$ from the fours column. This leaves $0$ in the fours column and makes the twos column $10_2$.
4. In the twos column, subtract: $10_2 – 1_2 = 1_2$. Write $1$ in the twos column of the answer.
5. In the fours column, the remaining digit is $0$. Subtract: $0 – 0 = 0$. Write $0$ in the fours column of the answer.

The final vertical arrangement is:

$$
\begin{array}{rccc}
& \overset{0}{\cancel{1}} & \overset{10}{\cancel{1}} & \overset{10}{0}_2 \\
– & 0 & 1 & 1_2 \\
\hline
& 0 & 1 & 1_2 \\
\hline
\end{array}
$$

Therefore, $110_2 – 011_2 = 11_2$ (or $011_2$).

Observe the place value chart below, which shows the step-by-step borrowing process for three-digit binary subtraction.

Practice Exercises

The following exercises are provided to test understanding of binary subtraction:

1. Subtract the following two-digit binary numbers:
   1. $11_2 – 10_2$
   2. $11_2 – 01_2$
   3. $10_2 – 01_2$
2. Subtract the following three-digit binary numbers:
   1. $111_2 – 010_2$
   2. $110_2 – 100_2$
   3. $101_2 – 011_2$
   4. $100_2 – 001_2$